Counting objects which behave like irreducible Brauer characters of finite groups
Let \(N\triangleleft G\), where G is a finite group. Suppose \(\theta\) is an irreducible character of N such that \(\theta (gng^{-1})=\theta (n)\), for all \(g\in G\), \(n\in N\). Define Irr(G\(| \theta)\) as the set of those \(\chi\in Irr(G)\) such that \(\theta\) is a constituent of \(\chi_ N\). Schur and Gallagher gave each a description of the cardinality of \(| Irr(G| \theta)|\). Namely, Gallagher did it as follows. Define \(\bar g\in G/N\) to be \(\theta\)-good if some extension of \(\theta\) to \(<N,g>\) is invariant in C, where \(C/N=C_{G/N}(\bar g)\). He shows that \(| Irr(G| \theta)| =\#\{conjugacy\) classes of G/N, consisting of \(\theta\)-good elements\(\}\). An analogous result holds for \(\theta\in IBr(N)\), where IBr(N) stands for the irreducible Brauer characters with respect to some prime p. Supposing \(\theta\) to be G-invariant, it holds that \(| IBr(G| \theta)|\) is equal to the number of \(\theta\)-good p-regular conjugacy classes in G/N. In this paper the above research area is also done for ``\(\theta\)-good \(\pi\)-classes'' where \(\pi\) is a set of primes. The real purpose of this paper is to prove an analog of Gallagher's theorem in case G is a \(\pi\)- separable group. Your reviewer had to be short here, so the interested reader is referred to the paper's introduction.
\(\theta \) -good p-regular conjugacy classes
scientific article; zbMATH DE number 4071171